Basic concepts on structural dynamics

The dynamic behavior of structures is an important topic in many fields. Aerospace engineers must understand dynamics to simulate space vehicles and airplanes, while mechanical engineers must understand dynamics to isolate or control the vibration of machinery. In civil engineering, an understanding of structural dynamics is important in the design and retrofit of structures to withstand severe dynamic loading from earthquakes, hurricanes, and strong winds, or to identify the occurrence and location of damage within an existing structure.

1. BASIC CONCEPTS ON STRUCTURAL DYNAMICS Dr.L.V . Prasad .M Department of Civil Engineering National Institute of Technology Silchar E-mail : [email protected] 21-Nov-16 1

2. What is Dynamics ? The word dynamic simply means “changes with time” 21-Nov-16 2 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

3. Basic difference between static and dynamic loading P P(t) Resistance due to internal elastic forces of structure Accelerations producing inertia forces (inertia forces form a significant portion of load equilibrated by the internal elastic forces of the structure) Static Dynamic 21-Nov-16 3 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS In static problem: Response due to static loading is displacement only. In dynamic problem: Response due to dynamic loading is displacement, velocity and acceleration.

4. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 4 Most Common Causes Dynamic Effect In The Structure Initial conditions : Initial conditions such as velocity and displacement produce dynamic effect in the system. Ex: Consider a lift moving up or down with an initial velocity. When the lift is suddenly stopped , the cabin begin to vibrate up and down since it posses initial velocity. Applied forces : Some times vibration in the system is produced due to application of external forces. Ex: i ) A building subjected to bomb blast or wind forces ii) Machine foundation. Support motions : Structures are often subjected to vibration due to influence of support motions . Ex: Earthquake motion.

5. Vibration and oscillation: If motion of the structure is oscillating (pendulum) or reciprocatory along with deformation of the structure, it is termed as VIBRATION. In case there is no deformation which implies only rigid body motion, it is termed as OSCILLATION. Free vibration: Vibration of a system which is initiated by a force which is subsequently withdrawn. Hence this vibration occurs without the external force. Forced Vibration: If the external force is also involved during vibration, then it is forced vibration. Basic Concepts of Structural dynamics 21-Nov-16 5 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

6. Damping: All real life structures, when subjected to vibration resist it. Due to this the amplitude of the vibration gradually, reduces with respect to time. In case of free vibration, the motion is damped out eventually. Damping forces depend on a number of factors and it is very difficult to quantify them. The commonly used representation is viscous damping wherein damping force is expressed as F d =C x . where x . = velocity and C=damping constant. Basic Concepts of Structural dynamics 21-Nov-16 6 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

7. The number of independent displacement components that must be considered to represent the effects of all significant inertia forces of a structure. Dynamic Degrees of Freedom Depending upon the co-ordinates to describe the motion, we have 1. Single degree of freedom system ( SDoF ). 2. Multiple degree of freedom ( MDoF ). 3. Continuous system. 21-Nov-16 7 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

8. Single Degree of Freedom: If a single coordinate is sufficient to define the position or geometry of the mass of the system at any instant of time is called single or one degree of freedom system. Multiple degree of freedom ( MDoF ): If more than one independent coordinate is required to completely specify the position or geometry of different masses of the system at any instant of time, is called multiple degrees of freedom system. Continuous system: If the mass of a system may be considered to be distributed over its entire length as shown in figure, in which the mass is considered to have infinite degrees of freedom, it is referred to as a continuous system. It is also known as distributed system. Dynamic Degrees of Freedom 21-Nov-16 8 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

9. Single Degree of Freedom Vertical translation Horizontal translation Horizontal translation Rotation 21-Nov-16 9 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

10. Degrees of freedom: –If more than one independent coordinate is required to completely specify the position or geometry of different masses of the system at any instant of time, is called multiple degrees of freedom system. Multiple Degrees of Freedom 21-Nov-16 10 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS Example for MDOF system

11. Continuous system: Degrees of freedom: –If the mass of a system may be considered to be distributed over its entire length as shown in figure, in which the mass is considered to have infinite degrees of freedom, it is referred to as a continuous system. It is also known as distributed system. –Example for continuous system: 21-Nov-16 11 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

12. Mathematical model - SDOF System Mass element ,m - representing the mass and inertial characteristic of the structure Spring element ,k - representing the elastic restoring force and potential energy capacity of the structure. Dashpot , c - representing the frictional characteristics and energy losses of the structure Excitation force, P(t) - represents the external force acting on structure. P(t) x m k c  F = m × x ·· = p ( t ) – cx · – kx mx ·· + cx · + kx = p ( t ) 21-Nov-16 12 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

13. Mathematical model - SDOF System Undamped (C =0 &P(t)=0) Free Vibration Damped ( C 0 &P(t)=0) Undamped (C =0 &P(t)  0) 2. Forced Vibration Damped ( C 0 &P(t)  0) 21-Nov-16 13 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

14. Equation of Motion - SDOF System 1.Simple Harmonic motion 2. Newtown’s Law of motion 3. Energy methods 4.Rayleights method 5.D’alembert’s method Differential equation describing the motion is known as equation of motion. 21-Nov-16 14 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

15. If the acceleration of a particle in a rectilinear motion is always proportional to the distance of the particle from a fixed point on the path and is directed towards the fixed point , then the particle is said to be in SHM. Simple Harmonic motion method : SHM is the simplest form of periodic motion. •In differential equation form, SHM is represented as 𝑥 ∝−𝑥 −−−(1) 21-Nov-16 15 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

16. Newton’s second law of motion : The rate of change of momentum is proportional to the impressed forces and takes place in the direction in which the force acts. Consider a spring – mass system of figure which is assumed to move only along the vertical direction. It has only one degree of freedom, because its motion is described by a single coordinate x. 21-Nov-16 Dr.L.V.Prasad, Assistant Professor, Civil Engineering Dept, NITS 16

17. Energy method: Conservative system: Total sum of energy is constant at all time. 21-Nov-16 17 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

18. Rayleigh’s method: Maximum K.E. at the equilibrium position is equal to the maximum potential energy at the extreme position. 21-Nov-16 18 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

19. D’Alembert’s method: D’Alemberts principle states that ‘a system may be in dynamic equilibrium by adding to the external forces, an imaginary force, which is commonly known as the inertia force’. Using D’Alembert’s principle, to bring the body to a dynamic equilibrium position, the inertia force ‘𝑚𝑥 is to be added in the direction opposite to the direction of motion. 21-Nov-16 19 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

20. P(t ) =0 x m k mx ·· + cx · + kx = p ( t ) Free Vibration of Undamped - SDOF System 21-Nov-16 20 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

21. Amplitude of motion t x v o or where, x X = initial displacement V = initial velocity  t V o = X . o &  =  Free Vibration of Undamped - SDOF System 21-Nov-16 21 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS  = p is called circular frequency or angular frequency of vibration ( Rad /s)

22. Free Vibration of damped SDOF systems (Dimensionless parameter ) - A where, x m k c 21-Nov-16 22 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS  is called circular frequency or angular frequency of vibration ( Rad /s)

23. Solution of Eq.(A) may be obtained by a function in the form x = e rt where r is a constant to be determined. Substituting this into (A) we obtain, In order for this equation to be valid for all values of t, or Free Vibration of damped SDOF systems 21-Nov-16 23 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

24. Thus and are solutions and, provided r 1 and r 2 are different from one another, the complete solution is The constants of integration c 1 and c 2 must be evaluated from the initial conditions of the motion. Note that for >1 , r 1 and r 2 are real and negative for <1 , r 1 and r 2 are imaginary and for =1 , r 1 = r 2 = -p Solution depends on whether is smaller than, greater than, or equal to one. Free Vibration of damped SDOF systems 21-Nov-16 24 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

25. For (Light Damping) : ‘ A’ and ‘B’ are related to the initial conditions as follows (B) In other words, Eqn.B can also be written as, where, Free Vibration of damped SDOF systems 21-Nov-16 25 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

26. Extremum point ( ) Point of tangency ( ) T d = 2 π / p d x n X n+1 t x 21-Nov-16 26 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

27. Such system is said to be over damped or super critically damped. i.e., the response equation will be sum of two exponentially decaying curve In this case r1 and r2 are real negative roots. For ( Heavy Damping) x o x o t 21-Nov-16 27 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

28. Such system is said to be critically damped. The value of ‘c’ for which Is known as the critical coefficient of damping With initial conditions , Therefore, For 21-Nov-16 28 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

29. Example 1 : A cantilever beam AB of length L is attached to a spring k and mass M as shown in Figure. ( i ) form the equation of motion and (ii) Find an expression for the frequency of motion. Stiffness due to applied mass M is 𝑘 𝑏 =𝑀/Δ=3𝐸𝐼/𝐿 3 Equivalent spring stiffness, 𝑘 𝑒 =𝑘 𝑏 +𝑘 𝑘 𝑒 =(3𝐸𝐼/𝐿 3 )+k 𝑘 𝑒 =(3𝐸𝐼+𝑘𝐿 3 )/𝐿 3 The differential equation of motion is, 𝑚𝑥 .. =−𝑘 𝑒 𝑥 The frequency of vibration, 21-Nov-16 29 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS

30. 21-Nov-16 Dr.L.V.Prasad, Assistant Professor, Civil Engineering Dept, NITS 30 Problem 2 : Calculate the natural angular frequency of the frame shown in figure. Compute also natural period of vibration. If the initial displacement is 25 mm and initial velocity is 25 mm/s what is the amplitude and displacement @t =1s. In this case, the restoring force in the form of spring force is provided by AB and CD which are columns. The equivalent stiffness is computed on the basis that the spring actions of the two columns are in parallel.

31. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 31 Problem 2

32. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 32 Problem 3: Following data are given for a vibrating system with viscous damping mass m=4.5 kg, stiffness k= 30 N/m and damping C=0.12 Ns/m. Determine the logarithmic decrement, ratio of any 2 successful amplitudes.

33. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 33 Multiple degree of freedom systems A multi degrees of freedom ( dof ) system is one, which requires two or more coordinates to describe its motion. These coordinates are called generalized coordinates when they are independent of each other and equal in number to the degrees of freedom of the system

34. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 34 Two degree of freedom systems

35. 21-Nov-16 Dr.L.V.Prasad, Assistant Professor, Civil Engineering Dept, NITS 35 Problem 4: A pedestal bridge platform is truss supported as shown in Fig. by neglecting the self weight of the truss , estimate the frequency of vibration of the truss by idealizing a simple spring-mass system. Assume that are of cross section and young's modulus are same for all members.

36. 21-Nov-16 Dr.L.V.Prasad , Assistant Professor, Civil Engineering Dept, NITS 36 Member Force (P) Unit force (p) Length (l) Ppl /AE AB L BC L CF W/2 1/2 L WL/4 FE W/2 1/2 L WL/4 DE W/2 1/2 L WL/4 AD W/2 1/2 L WL/4 BD + W/√2 +1/√2 √2L WL/√2 BF + W/√2 +1/√2 √2L WL/√2 BE L Problem 4

37. 21-Nov-16 Dr.L.V.Prasad, Assistant Professor, Civil Engineering Dept, NITS 37 THANK YOU